Question

what is the area of this figure
2 ft
5 ft
2 ft
3 ft
16 ft
4 ft
8 ft
8 ft
square feet

Explanation:

Step1: Analyze the figure's structure

The figure can be divided into three rectangles. Let's identify their dimensions.
First rectangle: width = 2 ft, height = 5 ft.
Second rectangle: width = 2 + 2 = 4 ft (wait, no, let's re - examine. Wait, the bottom rectangle has width 8 ft and height 8 ft? Wait, maybe a better way: the total height is 16 ft? Wait, no, the labels: there is a part with height 5 ft, then 3 ft, then 8 ft? Wait, maybe the figure is composed of three rectangles:
1. Top rectangle: width = 2 ft, height = 5 ft.
2. Middle rectangle: width = 2+2 = 4 ft? No, wait, looking at the horizontal segments: 2 ft, then 2 ft, then 4 ft, and the bottom width is 8 ft. So 2 + 2+4 = 8, which matches the bottom width. For the vertical segments: 5 ft, then 3 ft, then 8 ft, and 5 + 3+8 = 16, which matches the left - side height.
So rectangle 1: width = 2 ft, height = 5 ft. Area \(A_1=2\times5 = 10\) square feet.
Rectangle 2: width = 2 + 2=4 ft? Wait, no, the middle rectangle: the width should be 2 (from first) + 2? Wait, no, the horizontal length for the middle part: after the first 2 ft, there is a 2 ft segment, so the width of the middle rectangle is 2 + 2 = 4 ft? Wait, no, the height of the middle rectangle is 3 ft. So area \(A_2=(2 + 2)\times3=4\times3 = 12\) square feet? Wait, no, maybe the middle rectangle has width 2+2 = 4 ft? Wait, no, let's do it properly.
The bottom rectangle: width = 8 ft, height = 8 ft. Area \(A_3 = 8\times8=64\) square feet.
The middle rectangle: above the bottom one, width = 8 - 4=4 ft? Wait, no, the horizontal segments: 2 ft, 2 ft, 4 ft. So the middle rectangle (between the bottom and the top) has width = 2+2 = 4 ft? No, the height of the middle rectangle is 3 ft, and the width is 2 + 2=4 ft? Wait, no, let's use the method of adding areas by dividing the figure into three parts:
1. Top part: width = 2 ft, height = 5 ft. Area \(A_1=2\times5 = 10\).
2. Middle part: width = 2 + 2=4 ft, height = 3 ft. Area \(A_2 = 4\times3=12\).
3. Bottom part: width = 8 ft, height = 8 ft. Area \(A_3=8\times8 = 64\).
Wait, but 5+3 + 8=16, which matches the left - side height. And 2+2 + 4=8, which matches the bottom width.

Step2: Calculate the area of each rectangle

- Area of top rectangle (\(A_1\)): \(A_1=2\times5 = 10\) square feet.
- Area of middle rectangle (\(A_2\)): The width of the middle rectangle is \(2 + 2=4\) ft (since the first part is 2 ft, then another 2 ft), and height is 3 ft. So \(A_2 = 4\times3=12\) square feet.
- Area of bottom rectangle (\(A_3\)): Width is 8 ft, height is 8 ft. So \(A_3=8\times8 = 64\) square feet.

Step3: Sum up the areas

Total area \(A=A_1 + A_2+A_3=10 + 12+64=86\)? Wait, no, maybe my division is wrong. Wait, another way: the figure can be seen as a large rectangle minus some parts, but it's easier to add. Wait, maybe the correct division:
Wait, the bottom rectangle: width 8 ft, height 8 ft. Then above it, a rectangle with width 8 - 4=4 ft? No, wait, let's look at the horizontal lengths: 2 ft, 2 ft, 4 ft. So the top rectangle: 2 ft (width) x 5 ft (height). The middle rectangle: (2 + 2) ft (width) x 3 ft (height). The bottom rectangle: 8 ft (width) x 8 ft (height). Wait, but 2+2 + 4=8, and 5+3 + 8=16.
Wait, another approach: The total area can also be calculated as the area of the large rectangle (8 ft x 16 ft) minus the area of the missing parts. But the missing parts: on the right - hand side, there are two rectangles. Wait, no, maybe my initial division is wrong. Let's re - examine the labels:
Left - side height: 16 ft.
Bottom width: 8 ft.
Top - most segment: 2 ft (width), 5 ft (height).
Then a segment: 2 ft (width extension), 3 ft (height).
Then the bottom: 8 ft (width), 8 ft (height).
Wait, maybe the correct dimensions:
Rectangle 1: width = 2 ft, height = 5 ft. Area = 2*5 = 10.
Rectangle 2: width = 2+2 = 4 ft, height = 3 ft. Area = 4*3 = 12.
Rectangle 3: width = 8 ft, height = 8 ft. Area = 8*8 = 64.
Total area=10 + 12+64 = 86? Wait, no, 10+12 = 22, 22+64 = 86. But wait, let's check with another method.
Wait, maybe the height of the bottom rectangle is 8 ft, the middle part (between 5 ft and 8 ft) is 3 ft, and the top is 5 ft. The width of the top rectangle is 2 ft, the middle rectangle (above the bottom 8 ft x 8 ft) has width 2+2 = 4 ft (since the top is 2 ft, then a 2 ft extension), and the bottom is 8 ft.
Wait, but let's calculate the area by adding the three rectangles:
1. Top: 2*5 = 10.
2. Middle: (2 + 2)*3=4*3 = 12.
3. Bottom: 8*8 = 64.
10+12 + 64=86. Wait, but maybe I made a mistake in the middle rectangle's width. Wait, the middle rectangle: after the top 2 ft, there is a 2 ft, so the width of the middle rectangle is 2+2 = 4 ft, and the height is 3 ft. Then the bottom rectangle is 8 ft wide and 8 ft tall.
Wait, another way: Let's use the coordinates. Let's assume the bottom - left corner is (0,0). Then:
- Top rectangle: from x = 0 to x = 2, y = 11 to y = 16 (since 16 - 5=11). Area: 2*(16 - 11)=2*5 = 10.
- Middle rectangle: from x = 0 to x = 4 (2+2), y = 8 to y = 11 (11 - 3=8). Area: 4*(11 - 8)=4*3 = 12.
- Bottom rectangle: from x = 0 to x = 8, y = 0 to y = 8. Area: 8*8 = 64.
Total area: 10+12 + 64=86. Wait, but let's check again. Wait, maybe the height of the bottom rectangle is 8 ft, the middle is 3 ft, and the top is 5 ft, and 5+3 + 8=16, which matches the left - side height. The width of the top is 2 ft, middle is 2+2 = 4 ft, bottom is 8 ft, and 2+2 + 4=8, which matches the bottom width. So the calculation seems correct.

Answer:

86